Étienne Montucla's father was a merchant who lived and worked in Lyons. It was in that town that his son Étienne was born, was brought up, and began his education.
Étienne was educated in the Jesuit College in Lyon but, in 1741, while he was studying there his father died and he was left an orphan. At this stage his grandmother took over his upbringing. Étienne was interested in a wide range of studies at the Jesuit College. Riddle writes in :-
His attention was chiefly directed to the ancient classics; but having a natural taste for philosophical studies, and a powerful memory, he was enabled to acquire an accurate knowledge of several modern languages; among which Italian, German, Dutch, and English are mentioned. Under Le Père Bêraud, who was subsequently the tutor of Lalande, he made considerable proficiency in the study of mathematics and physics.
In 1745 Montucla's grandmother died and at this point he went to Toulouse where he began to study law. After completing his studies at Toulouse, he decided to go to Paris which would be the most suitable place to gain further legal training. There he became friends with Diderot, Lalande and d'Alembert who he met at the scientific soirées organised by Charles Antoine Jombert. He made many other friends among the top mathematicians of his day, and he began to combine his many interests and study the history of mathematics. Making the acquaintance of Jombert had other advantages, for he was a bookseller and publisher. The advice from Jombert, d'Alembert and Diderot set Montucla on the road of publishing on the history of mathematics.
His first work was Histoire des recherches sur la quadrature du cercle which was published by Jombert in Paris in 1754, and the high regard in which this work was held can be judged by the fact that on the strength of this work alone Montucla was elected a corresponding member of the Berlin Academy. Although the history of the problem of squaring the circle has changed greatly since Montucla's day this book is still a classic of its kind. Already at this time he was working on his masterpiece, Histoire des mathématiques, and in 1754 he announced that it would be published at a future date. He was undertaking a very major task, one which had never really been undertaken in this form before, and four years would pass before publication. Montucla needed an income to allow him to pursue his research interests and around this time he worked at the Gazette de Paris.
In 1756 Montucla published a work which shows the breadth of his scientific interests. A collaboration with the physician Joseph Morisot-Deslandes led to the publication of Recueil de pièces concernant l'inoculation de la petite vérole et propres à en prouver la sécurité et l'utilité, which provided a collection of sources on smallpox vaccination. Then in 1758 he published his two volume work, again with Jombert as his publisher, Histoire des mathématiques. The first volume covered the history of mathematics from ancient times to 1700, while the second volume was entirely devoted to 17th century mathematics. Vogel  describes Montucla's treatise as:-
... the first attempt of a history of mathematical ideas and problems. [Earlier works being mostly] lists of names, titles and dates ...
Montucla had been intending to produce a third volume covering the first half of the 18th century but the volume of work which had appeared during this time, and the difficulties of putting recent work into its historical context, led him to give up this aim. These two volumes rightly led to Montucla gaining a high reputation in Paris and he began to work for the French government.
In 1761 he went to Grenoble where he was appointed to the position of intendant to Dauphiné. Dauphiné was a region of south-eastern France which had originally been independent. In the 14th century it had been sold to Charles V of France who then ceded Dauphiné to his heir, a practice which later Kings continued. A Parlement of Dauphiné was created in 1453 by the son of Charles VII who used his position there to cause an argument with his father, who then responded by annexing Dauphiné to France. In 1628 the French government appointed an intendant to Dauphiné to prevent any notions of independence it might have. The intendant was an important royal official with control over the area, and it was to that position that Montucla was appointed in 1761. He was based in Grenoble, the principal town of Dauphiné. There he met Marie Françoise Romand and they married in 1763.
A year after his marriage the French government sent Montucla to Cayenne on the north-eastern coast of South America. Jean Richer had been sent there by the French government almost exactly 100 years earlier and Montucla also went there as an astronomer. In 1643 French merchants founded a town on the site of the present day city of Cayenne, a port of French Guiana, calling the settlement La Ravardière. The Indians later destroyed the town but the French reoccupied it in 1664 and were awarded French Guiana in a treaty of 1667, four years before they sent Jean Richer there. In the 1700s the French worked hard at building up their South American territory. Montucla was appointed royal astronomer and secretary to a French expedition to French Guiana in 1764 and was sent to La Ravardière. After spending over a year he returned to France in 1765. We note in passing that La Ravardière was renamed Cayenne in 1777.
After his return to France, Montucla received another major government appointment, as Head Clerk of Royal Buildings, holding the post from 1766 to 1789. He was also appointed as Royal Censor of mathematical works in 1775. A few years later he published another text for which he is famed, namely a new edition of Ozanam's Récréations mathématiques et physiques in 1778. Riddle writes in  that Montucla:-
... so greatly enlarged and improved the "Recreations" of Ozanam, that he may be said to have made the work his own. ... so carefully had he concealed his connection with the work, that, on its completion, a copy of it was sent to him, in his capacity of censor, for examination and approval. Besides expunging from the work of Ozanam much that was absurd, puerile, and obsolete, he enriched his edition with dissertations upon almost every branch of practical science ...
His edition was influential in popularising geometric dissection problems which involve the cutting of geometric figures into pieces that can be arranged to form other geometric figures. Charles Hutton translated Montucla's edition of Ozanam's Récréations mathématiques et physiques into English in 1803 and Riddle's edition was published in 1844, called Recreations in science and natural philosophy.
In 1788 Fourier sent an algebra paper to Montucla for his opinion. Fourier, however, did not receive an opinion and wrote in a letter a year later:-
I begin to take M Montucla at his word when he tells us he has fallen out with learned analysis: I wait calmly for him to be reconciled with it.
This may mean that Montucla had stopped doing mathematics at this time but Fourier's comment is not properly understood.
Not surprisingly Montucla's royal posts vanished when the French Revolution began in 1789. By this time he was quite well off, having held high office for many years, but the Revolution saw him lose most of his wealth, as Riddle writes:-
... throwing him on the world in his old age, stripped of everything but his integrity, and the love and respect of his friends.
At this time he retired and went to live at Versailles where he worked on history of mathematics texts. He was, however, given further public appointments in 1795 even although he was seventy years old by this time. He was given the task of evaluating the treaties in the Ministry of Foreign Affairs, but was paid so little he lost financially. He also held a position in the office of the National Lottery.
Montucla's friends persuaded him to work on a new edition of his famous Histoire des mathématiques. In August 1799 Montucla published new editions through Agasse in Paris of the two volumes originally published in 1758. Montucla extensively revised and enlarged the two volumes. He had intended to extend his cover of history to the end of the 18th century and part of the third volume on this topic was printed by the time he died, four months after the publication of the new editions of 1799. Lalande, with the help of some other scientists, completed volumes three and four to give the coverage that Montucla had intended. Volume three covered 18th century pure mathematics, optics and mechanics in 832 pages, while the fourth volume covered 18th century astronomy, mathematical geography and navigation in 688 pages. Vogel writes in :-
Montucla's major work, the first classical history of mathematics, was a comprehensive and, relative to the state of contemporary scholarship, accurate description of the development of the subject in various countries. the account also included mechanics, astronomy, optics, and music ...
Sarton  wrote:-
It is still a very valuable book, though only for the wary 'connoisseur'; novices will do well to leave it alone.
The introduction to a reprint of the four volumes of Histoire des mathématiques in 1968 gives this assessment of Montucla's contribution:-
Montucla has given us a double lesson in the history of mathematics. Of these lessons, the first is what he has composed for us. The second is what he has written without intending to, that which, through his personality, se trouve être l'image des idées, des manières de voir de son époque et du milieu scientifique qu'il fréquentait et connaissait très bien.
Article by: J J O'Connor and E F Robertson